Course Outline (eventually including Assignments) and Some Supplementary Class Notes

For the most part we shall follow closely the appropriate sections of the reference(s) listed above.

• Preliminaries

• Three Notions of Smallness for subsets of the Reals
  
• Countable, Meager (1st Category), and Null (measure zero)
  
• The Reals are "not small", specifically they are neither countable, meager, or null (Theorems of Cantor, Baire, and Borel respectively)
  

• Discontinuities and Non-Differentiability (discussion only, no proofs)
  
• F-sigma sets, first class functions, and Lebesgue's Criterion for Riemann Integrability
• Nowhere differentiable functions and Lebesgue's Theorem on the differentiablity of monotone functions

• Review of Uniform Convergence (in Math 8105)
  

Homework 1    (Due Tuesday August 30)

• Lebesgue Measurable Sets and Functions
  

• Introduction
• Recalling the Riemann integral and its limitations
• Can we assign a "measure" to all subsets of the real line?

• Lebesgue outer measure   (Sections 1.1 and 1.2 in Stein)
  
• Preliminaries (decomposition theorems for open sets)
• Properties of Lebesgue outer measure

• Lebesgue measure   (Section 1.3 in Stein)
  
• Measurable sets form a sigma-algebra
• Closed sets are measurable (only using that countable unions of measurable sets are measurable)
  
• Different characterizations of Lebesgue measurability (using closed sets, G-delta sets and F-sigma sets)
• Countable additivity and the briefest of introductions to abstract measure spaces
  
• Continuity from above and below
• Translation invariance
• Steinhaus' theorem

Homework 2    (Due Tuesday September 13)

• Lebesgue measurable functions  (Section 2.1 in Folland)
• Equivalent definitions and the useful characterization using open sets
• Compositions and closure under algebraic and limiting operations
• Approximation by simple functions (and step functions)

• Littlewood's Three Principles  (Section 1.4 in Stein)      [Postponed for now]
• Every measurable set is nearly a "very nice set", such as a finite union of cubes or an open set
  
• The Lebesgue Density Theorem is arguably one of the strongest realizations of this principle 
• Every measurable function is nearly a continuous function (Lusin's theorem)
• Every convergent sequence of measurable functions is nearly uniformly convergent (Egorov's theorem)

Homework 3    (Due Thursday September 22)

• Lebesgue Integral

• Integration of non-negative measurable functions  (Section 2.2 in Folland)
• Simple functions and their integral (including properties such as monotonicity and linearity)
  
• Extension to all non-negative measurable functions (establishing linearity using MCT below)
• Monotone Convergence Theorem (proved using "continuity from below" for measures defined by simple functions)
  
• Chebyshev's inequality and proof that "f equals 0 a.e. if and only if the integral of f equals 0"
• Convergence Theorems for non-negative measurable functions and examples
• (Modified) Monotone Convergence Theorem and Fatou's Lemma

• Integration of extended real-valued and complex-valued measurable functions  (Section 2.3 in Folland)
  
• Space of all complex-valued integrable functions form a complex vector space on which the Lebesgue integral is a complex linear functional
  
• Dominated Convergence Theorem
• Discussion on how to establish the continuity and differentiablity of functions defined by integrals

Homework 4    (Due Thursday October 6)

• Definition of L^1, Completeness, and interchanging sums and integrals  (Section 2.3 in Folland continued, see also Section 2.4 in Folland and Section 2.2 in Stein)
  
• Examples of different modes of convergence
• Every sequence in L^1 which converges in norm contains a subsequence that converge almost everywhere
• Every absolutely convergent series in L^1 converges almost everywhere (and in L^1) to an L^1 function
• Conclusion of proof that L^1 is a complete normed space (a Banach space)
• Discussion that more generally a normed vector space X is complete if and only every absolutely convergent series in X converges

• Further Properties of the Lebesgue Integral
  
• Relationship with Riemann integration  (Theorem 2.28 in Folland)
  
• Absolute continuity of the Lebesgue integral and "small tails property"  (using the MCT as in proof of Proposition 1.12 in Section 2.1 of Stein)
  
• Translation and dilation invariance of the Lebesgue integral  (see  page 73 of Stein)

• An Approximation Theorem  (Section 2.2 in Stein, see also Section 2.3 and 2.6 in Folland)
  
• Simple functions and Continuous functions with compact support are both dense in L^1 -- another realization of Littlewood's second principle
  
• Continuity in L^1   (Proposition 2.5 in Section 2.2 of Stein)
  
• Short proofs of both the absolute continuity and "small tails property" of the Lebesgue integral

Exam 1   (In class on Thursday the 13th of October)

Old Exams for practice: 2021 (75 minutes), 2019 (75 minutes),  2018 (60 minutes),  2014 (60 minutes), and 2013 (75 minutes)

• Repeated Integration

• Fubini-Tonelli Theorem(s)  (Section 2.3)
• Statement of theorems and discussion of examples
• Appendix on Measurability of "cylinder sets and functions"
  
• Proof of Fubini's theorem and Tonelli's theorem

Homework 5    (Due Thursday October 27)

• Applications
  
• Lebesgue integral under linear transformations (example of an "it suffices to consider Borel measurable functions" argument) 
• Convolutions: 
  
• Basic Properties
• Approximations to the identity in L^1 (and in uniform norm for function which are bounded and uniformly continuous)
  
• Application of L^1 result: Smooth functions with compact support are dense in L^1
• Application of uniform norm result: Weierstrass approximation theorem
• Introduction to the Fourier transform on R^n and proof of the Fourier inversion formula
• Informal discussion on PDEs: Heat diffusion, Dirichlet problem for the upper-half plane, wave equation and the Cauchy problem

Homework 6    (Due Tuesday November 8)

• Some Functional Analysis
  

• Hilbert Spaces
• Inner product spaces and Hilbert spaces
• Orthonormal sets and Fourier series
• Existence of a basis (for separable spaces this follows from Gram-Schmidt, in the non-separable case one requires the axiom of choice)
  
• Construction of an orthonormal basis for L^2(0,1) and the (periodic) Weierstrass approximation theorem on C([0,1]) [Statements only - Proof sketched in Problem Session]
• Handout on Fourier Series  [non-examinable]
• Closed subspaces, orthogonal projections and the Riesz Representation Theorem (for Hilbert Spaces)

 Homework 7    (Due Thursday the 17th of November)

• Linear Functionals and Dual spaces (for general normed vector spaces)

• Basic Theory of L^p Spaces
  
• Holder's inequality and Minkowski's inequality
  
• Dense subclasses [Proof will not be covered in class and are non-examinable]
  
• ** Qualitative and quantitative relationships between the different L^p space (applications of Holder) [Statements and proofs will not be covered in class and are non-examinable]
• The Dual Space of L^p (including a sketch proof of the Riesz Representation Theorem for L^p functions)  ** We gave a different examinable approach to the RRT for finite measure spaces when p=1.
  
• "Converse of Holder"      ** We only discussed "Part (i) for q=1 (which was p=1 for us)"

Homework 8    (Due Thursday the 2nd of December)